This paper presents new models for predicting the box compression strength (BCT) of solid board boxes. The models account for phenomena that are significant in packaging logistics under various atmospheric conditions, in particular relative humidity. They were developed on the basis of experimental studies involving top-to-bottom compression tests of boxes made from three types of solid board supplied by three different manufacturers, at relative humidity levels of 30% RH, 50% RH, and 80% RH. The aim of the study was to develop a semi-empirical mathematical modelling method for the compression strength of solid board boxes and to predict the strength of solid board packaging at any given relative humidity. The developed equation was used to simulate the box strength under the conditions specified in the experiment, and the relative error was calculated. The material constants were determined using the short-span compression test (SCT) of solid board and the Young’s modulus, in accordance with ISO standards. The obtained results indicate that humidity plays a significant role in the deformation process of the box and should be taken into account in mathematical modelling of packaging strength.